Problem: Calculate the gravitational acceleration on Earth.
The force on an object of mass \(m_1\) near the surface of the Earth is
\[\begin{aligned} F = m_1 g\end{aligned}\]
where \(g\) is the gravitational acceleration that we are interested in. Newton’s gravity formula is given as below
\[\begin{aligned} F = m_1 g = G\frac{m_1 m_2}{r^2}\end{aligned}\]
where \(m_2\) is the mass of the Earth, \(r\) is the radius of the Earth, and \(G\) is the gravitational constant. Each value is given as below
\(G = 6.67\times 10^{-11} \left[\frac{m^3}{kg s^2}\right]\)
\(m_{Earth}=m_2 = 5.97 \times 10^{24} [kg]\)
\(r_{Earth}=r = 6371 [km]\)
Substitute all the values
\[\begin{aligned} F = m_1 g &= G\frac{m_1 m_2}{r^2}\\ g &= G \frac{m_2}{r^2} = 6.67\times 10^{-11}\frac{5.97 \times 10^{24}}{ 6371^2 \times 1000^2 \left[ \frac{m}{km}\right]}=9.81 \left[ \frac{m}{s^2}\right]\end{aligned}\]
